1. Field of the Invention
The present invention relates to a method for analyzing an electromagnetic field, which is useful in design of an electronic or electric equipment (e.g., a transformer, a reactor, or a motor) utilizing an electromagnetic phenomenon.
2. Description of the Related Art
The following series of equations or equivalent equations obtained by transforming the equations with another variable are used as fundamental equations for an electromagnetic field analysis. EQU .gradient..times.H=J.sub.0 +J.sub.e ( 1) EQU B=.gradient..times.A (2) ##EQU1## EQU B=.mu.H (4) EQU J.sub.e =.rho.E (5)
where each letter or symbol with an arrow (.fwdarw.) represents a vector having components in three directions in a space, A is the vector potential [V.multidot.s/m], .phi. is the scalar potential [V], J.sub.0 is the exciting current density [A/m.sup.2 ], J.sub.e is the eddy current density [A/m.sup.2 ], .rho. is the electric conductivity [S/m], and .mu. is the permeability [H/m].
To solve these equations in practice, techniques such as a finite-element method are used. According to the finite-element method, a region of interest is spatially divided into a large number of sub-regions each having a relatively simple shape called "element", and a target unknown function is approximated by a relatively simple function on each element. More specifically, a potential to be obtained is developed with an approximate function on each element, and the potential is then obtained so as to minimize the energy within the region by using the variation principle. An approximate method using an approximate function for giving an unknown variable 2 on an edge of each element 1 is called edge-element approximation, as shown in FIG. 2. FIG. 2 exemplifies a hexahedral element 1. However, another edge-element approximation equally applies to an element having any other shape. An electromagnetic field analysis using an edge-element approximation is described in, e.g., "A. KAMEARI: Three Dimensional Eddy Current Calculation Using Edge Elements for Magnetic Vector Potential; Applied Electromagnetics in Materials: pp. 225-236, 1988".
An electromagnetic field analysis using an edge-element approximate function will be described below. The approximate function used in an edge-element approximation belongs to a function space in which the gauge transformation for electromagnetic field is held, as described in "M. Hano: a finite-element analysis of a three-dimensional electromagnetic field and its gauge conditions; Materials of the society for the study of electromagnetic theory; EMT-89-45, 1989". For this reason, a gauge for nullifying the scalar potential .phi. in equation (3) can be employed. Therefore, .phi.=0 is assumed in the following description. A discrete formation of the fundamental equations for electromagnetic field is made by using an edge-element approximate function in accordance with the Galerkin's method equivalent to the variation principle. The vector potential in the element is approximated as follows: ##EQU2## where N.sub.q is the edge-element approximate function (vector function), A.sub.q is the unknown variable, the suffix q or suffix k (to be described later) represents an edge number, and m represents the total number of edges of each element. The Galerkin's method is applied to equation (1). That is, N.sub.k is multiplied with equation (1) to perform the volume integration within the element, the thus-obtained integral equations for all the elements are added to each other, and the resulting equation is rearranged with the Gauss' theorem and the vector formulas to yield the following equation: ##EQU3## where ##EQU4## is the integration within the element, ##EQU5## is the sum for all the elements, n is the unit vector normal to the boundary plane, and ##EQU6## is the surface integration on the element surface.
The above surface integral term is generally set zero. In that case, unless a boundary condition is specified on a boundary outside the region of interest, a solution for nullifying the surface integral term can be obtained. Let the surface integral term be zero in the following description. When equation (6) is substituted into equation (7), and, for example, a backward difference is applied to the time differential term, simultaneous linear equations having A.sub.a as unknown variable and an integral term as coefficient are obtained. ##EQU7## where .DELTA.t is the time difference, and A.sub.q is the A.sub.q value .DELTA.t before. In this case, the time differential term is represented by difference approximation, but complex approximation (j.omega. method: j is the imaginary unit, and .omega. is the angular frequency) can be used alternatively.
When the above discreted equation is applied to the region of interest, the electromagnetic field generated in the region can be analyzed. That is, equation (8) is calculated on the basis of input data such as shape data, physical property data, time data, boundary condition data, and the exciting current density J.sub.0 applied to the electric conductor included in the region, all of which are input as analysis initial conditions. At this time, the integrations of equation (8) are calculated by, e.g., numerical integration. When the integrations are executed to obtain simultaneous linear equations for the respective elements and they are added for all the elements, comprehensive simultaneous linear equations are obtained. These are solved by using the Gauss' elimination method as a direct method, the incomplete Cholesky conjugate gradient method as an iteration method, or the like, and finally the target potential can be calculated. The Gauss' elimination method, the incomplete Cholesky conjugate gradient method, and the like are described in, e.g., "Murata, Oguni, and Karaki: Supercomputer; 1985". The edge-element approximation has the highest accuracy at a high speed among the electromagnetic field analysis techniques because the edge-element approximation is a physically accurate approximation. Therefore, the edge-element approximation is very effective for an electromagnetic field analysis in design of an electronic or electric equipment.
In the electromagnetic field analysis described above, the exciting current density J.sub.0 is a quantity which must be given as an initial condition for analysis. It must satisfy the condition of continuity of current of the next equation required by the Maxwell's equations: EQU .gradient..multidot.J.sub.0 =0 (9)
In a conventional electromagnetic field analysis using an edge-element approximation, if the exciting current density J.sub.0 given does not satisfy the condition of continuity of current, no convergence can be attained in the incomplete Cholesky conjugate gradient method which is generally used for solving simultaneous linear equations. Thus, the electromagnetic field cannot be calculated. For example, as reported in "Fujiwara: Scripts for lectures of the second seminar on numerical analysis of electromagnetic field; p. 7, 1991", when a region including a rectangular electric conductor is divided into elements of rectangular parallelopipeds, and data which do not satisfy the condition of continuity of current of equation (9) in the electric conductor are input to a computer to perform the analysis, no convergence can be obtained in the incomplete Cholesky conjugate gradient method. That is, in a conventional analysis technique, to accurately analyze an electromagnetic field generated in a region including an electric conductor, an exciting current density J.sub.0 which is a vector quantity and meets the condition of continuity of current in equation (9) must be given. If the shape of the electric conductor is simple, the exciting current density J.sub.0 to be given can be relatively easily determined so as to meet the condition of continuity of current, and it can be input to the computer. However, when the shape of the electric conductor or the element divisions are complicated, the exciting current density J.sub.0 cannot be easily determined in order to meet the condition of continuity of current. It is then impossible to input an initial condition of the exciting current density J.sub.0 to the computer.
After all, in a conventional electromagnetic field analysis method, to analyze an electromagnetic field of a region including a relatively complicated electric conductor structure, an exciting current density J.sub.0 which is a vector quantity and meets the condition of continuity of current within the electric conductor cannot be input as an initial condition in a computer. As a result, the electromagnetic field cannot be analyzed with high accuracy.